The order-theory tag has no usage guidance.

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588 views

### Is it reasonable to define `poset homotopy' as a `natural transformation of posets'?

Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, all topological ...

**9**

votes

**4**answers

864 views

### Universal order type

Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear ...

**2**

votes

**0**answers

78 views

### A specific notion between the notions of transversal and system of distinct representatives.

Let $X$ be a set, let $\mathcal{C}$ be a collection of subsets of $X$, and let $x_1, \dots , x_k \in X$. Say that the sequence $\{x_i\}_{i=1\dots k}$ is a sequential transversal (of length $k$) ...

**0**

votes

**2**answers

358 views

### Semilattices with n elements

How many n-element semilattices there are?
For example, for n-element partially ordered set we can figured out, that there are $2^{n*(n-1)}$ possible sets.
And can I find all possible n-element ...

**3**

votes

**1**answer

896 views

### Well ordering of countably branching well founded trees

Hello everybody,
I would like to say, before stating the question, that I'm not a mathematician, and therefore i apologize in advance if the question is trivial, or well-known. I'll try to state it ...

**8**

votes

**1**answer

392 views

### Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...

**1**

vote

**1**answer

323 views

### Modern books about orders and algebras on trees [closed]

Please help to find books about orders and algebras on trees.
If there is no modern books, please advice good old ones!
I'm more interested in finite trees (my current problem), but infinite ones are ...

**8**

votes

**1**answer

338 views

### Decomposing posets into countably many chains

A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some ...

**2**

votes

**2**answers

199 views

### Of what kind of complemented bounded poset are the structures in my quasi-variety?

I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far:
Let
$\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$
be the structure with ...

**7**

votes

**2**answers

633 views

### Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...

**2**

votes

**2**answers

526 views

### A problem about posets similar to Suslin's problem

Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...

**3**

votes

**2**answers

734 views

### Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (here we allow equality in linear extensions) for $A, B$ are
$$A_L=\{A_1=\{a< b< ...

**0**

votes

**1**answer

206 views

### Is every countable well-order embeddable in \mathbb{R}? [duplicate]

Possible Duplicate:
Order types of positive reals
The title is self-explanatory.

**3**

votes

**2**answers

241 views

### What do you call a lattice whose meet operation preserves disjointness of subsets?

To make my question more precise and compact (and probably more intuitive), let me define the following:
A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x ...

**1**

vote

**0**answers

327 views

### Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...

**10**

votes

**1**answer

456 views

### Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have ...

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votes

**2**answers

292 views

### orders and length functions on finitely generated groups

Let $G$ be a finitely generated group with the natural word length function ($|x|$ is the length of the shortest word in generators of $G$ representing $x$). We call a partial left invariant order ...

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**0**answers

271 views

### Higher-order dimension in posets: a reference request

Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...

**0**

votes

**2**answers

297 views

### Is this a pre-ordered commutative semigroup?

Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...

**3**

votes

**0**answers

383 views

### a poset with small “cycles”

(a followup to this recent question)
I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...):
Suppose that $z$ is covered by ...

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**3**answers

3k views

### Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...

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**4**answers

1k views

### What are examples of ordered fields that do not have the Archimedean Property?

Are the computable numbers one example?

**3**

votes

**2**answers

441 views

### Arithmetic of ordered sets more general than ordinals

Motivation. Having read about infinite time Turing machines and ω-languages, I was thinking about more general notions of languages and “computation time”. Languages over strings of length ...

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2k views

### A principle of mathematical induction for partially ordered sets with infima?

Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old ...

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1k views

### When does a Galois connection induce a topology?

Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps
$\Phi: X \rightarrow Y, \ \Psi: Y ...

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votes

**1**answer

213 views

### not sure what these basic FO symbols mean [closed]

I know some basic logic symbols, but i'm not sure what this formula means:
Fragment:
http://i35.tinypic.com/14jt1n9.png
Full paper:
http://www.newton.ac.uk/preprints/NI07003.pdf
in particular, what ...

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votes

**0**answers

195 views

### An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows.
Objects: Pairs ($X$, $\tau : X ...

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votes

**2**answers

435 views

### Fraissé limit of the finite linear orderings

Hodges in his Shorter Model Theory promises to show "in what sense the finite linear orderings 'tend to' the rationals rather than, say, the ordering of the integers" (p. 160). After going through his ...

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vote

**1**answer

567 views

### The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?

Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa(P)$ denote the poset of $\kappa$-small subposets of $P$ and let ...

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votes

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199 views

### Finite categories and partial orders

I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theory is related to ...

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**0**answers

180 views

### How linearly independent are the obvious combinatorial invariants of a Bruhat interval?

Let $[u, v]$ be a Bruhat interval in some Coxeter group. Let $I$ be the set of all Bruhat intervals. I am interested in functions $I \to \mathbb{Z}$ which are invariant under poset isomorphisms. ...

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votes

**1**answer

293 views

### Monotonic maximal chains in a Coxeter group

Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in ...

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**1**answer

528 views

### Any further applications of Freudenthal's 1936 Spectral Theorem?

Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...

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862 views

### Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, ...

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**0**answers

177 views

### Vector-valued valuations on lattices

There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x \wedge y) + v(x ...

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votes

**5**answers

1k views

### Explicit ordering on set with larger cardinality than R

Is it possible to construct (without using Axoim of Choice) a totally ordered set S with cardinality larger than $\mathbb{R}$?
Motivation: A total ordering is often called a “linear ordering”. I have ...

**28**

votes

**2**answers

689 views

### What is the minimal size of a partial order that is universal for all partial orders of size n?

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds
order-preservingly into $\mathbb{B}$.
For example, every partial order
...

**3**

votes

**2**answers

1k views

### Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?

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votes

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828 views

### Order types of positive reals

Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the ...

**6**

votes

**1**answer

869 views

### Correspondence between functions on a set and “states” on its power set

Let $L$ be the poset (ordered by set inclusion) that is the power set of some set $X$.
A state is a function $s:L \rightarrow [0,1]$ satisfying
i) for {$p_1,p_2,...$}, $p_i \in L$ a
pairwise ...

**-2**

votes

**1**answer

645 views

### Terminology: Lexicographical order [closed]

I would like to order a group of things by a set of rules of decreasing precedence. Please critique this sentence to help illustrate that:
We define a lexicographical ordering for sheep by looking at ...

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votes

**4**answers

479 views

### (a,b) ≤ (a',b') iff b ≤ a' or (b' = b and a ≤ a') where a≤b and a'≤b'. Has any one seen this order?

I have been mucking round with orders, and this is the order that I found I needed. I would like to know if it is defined somewhere else, it is kind of difficult to search for such things.
*EDIT*
...

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726 views

### Does the exact pair phenomenon for partial orders occur in your area of mathematics?

Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if
...

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votes

**9**answers

3k views

### How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...

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votes

**5**answers

3k views

### infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...

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votes

**1**answer

840 views

### Kuratowski closure-complement problem for other mathematical objects?

The original Kuratowski closure-complement problem asks:
How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a ...

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votes

**3**answers

1k views

### Constructing a metric over a lattice

Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$).
$f$ is said to be ...

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votes

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649 views

### Weak partitioning vs. strong partitioning

Let $U$ is a complete lattice with least element 0.
Weak partitioning is a collection $S$ of nonempty subsets of $U$ such that $\forall x\in S: x\cap\bigcup(S\setminus\{x\})=0$.
Strong partitioning ...

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votes

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658 views

### Filter-closed vs. chain-closed

Let A is a complete lattice.
I call a subset $S$ of A filter-closed when for every filter base $T$ in $S$ we have $\bigcap T\in S$. (A filter base is a nonempty, down directed set.)
I call a subset ...

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votes

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224 views

### Name for “lower/upper bounds” of arbitrary relations?

Given a partial order $R_{\leq}$ over a set $D$, the set of upper bounds under $R$ of a subset $S$ of $D$ is commonly defined as $\{ y \in D | \ \forall x\in S, x R y \}$.
(The set of lower bounds of ...